Problem: Nina made two investments: Investment $\text{A}$ has a value of $\$50$ at the end of the first year and increases by $8\%$ per year. Investment $\text{B}$ has a value of $\$60$ at the end of the first year and increases by $\$3$ per year. Nina checks the value of her investments once a year, at the end of the year. What is the first year in which Nina sees that investment $\text{A}$ 's value exceeded investment $\text{B}$ 's value?
Solution: Notice that investment $\text{A}$ 's value grows exponentially while investment $\text{B}$ 's value grows linearly. This means investment $\text{A}$ 's value is bound to exceed investment $\text{B}$ 's value at some point. Let's start calculating the value of each investment to see when that happens. Year Investment $\text{A}$ Investment $\text{B}$ (Multiply by $1.08$ each year.) (Add $3$ each year.) $1$ $\$50$ $\$60$ $2$ $\$54$ $\$63$ $3$ $\$58.32$ $\$66$ $4$ $\$62.99$ $\$69$ $5$ $\$68.02$ $\$72$ $6$ $\$73.47$ $\$75$ $7$ $\$79.34$ $\$78$ In conclusion, investment $\text{A}$ 's value will first exceed investment $\text{B}$ 's value in year number $7$.